We investigate the structure of k-positivity and Schmidt numbers for classes of linear maps and bipartite quantum states exhibiting symplectic group symmetry. Specifically, we consider linear maps on M_d(C) which are covariant under conjugation by unitary symplectic matrices S, and bipartite states which are invariant under S ⊗ S or S ⊗ \bar{S} actions, each parametrized by two real variables. We provide a complete characterization of all k-positivity and decomposability conditions for these maps and explicitly compute the Schmidt numbers for the corresponding bipartite states. In particular, our analysis yields a broad class of PPT states with Schmidt number d/2 and the first explicit constructions of (optimal) (d/2−1)-positive indecomposable linear maps for arbitrary d, achieving the best-known bounds. Overall, our results offer a natural and analytically tractable framework in which both strong forms of positive indecomposability and high degrees of PPT entanglement can be studied systematically. Finally, we show that the PPT squared conjecture holds true within the classes of PPT linear maps which are either symplectic covariant or conjugate-symplectic covariant.